Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1

Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1

Limits and Derivatives Exercise 13.1 Solutions

1. Evaluate the Given limit  (x + 3) 

Solution

 (x + 3) = 3 + 3 = 6


2. Evaluate the Given limit :  (x - 22/7) 

Solution


3. Evaluate the Given limit :   πr2.
Solution
πr2 = π(1)2 = π


4. Evaluate the Given limit :  (4x + 3)/(x - 2)
Solution

5. Evaluate the Given limit :  (x10 + x5 + 1)/(x – 1) 
Solution

6. Evaluate the Given limit :  [(x + 1)5 - 1]/x 
Solution

7. Evaluate the Given limit :   (3x2 - x - 10)/(x2 - 4) 
Solution
At x = 2, the value of the given rational function takes the form 0/0 . 

8. Evaluate the Given limit  (x4 -81)/(2x2 - 5x - 3) 
Solution
At x = 2, the value of the given rational function takes the form 0/0 .

9. Evaluate the Given limit :  (ax + b)/(cx + 1) 
Solution

10. Evaluate the Given limit :  
Solution

At z = 1, the value of the given function takes the form 0/0 . 
Put z1/6 = x so that z →1 as x →1.

11. Evaluate the Given limit (ax2 + bx + c)/(cx2 + bx + a) , a + b + c ≠ 0 .
Solution

= 1 [a + b + c ≠ 0 ] 

12. Evaluate the Given limit :  
Solution

At x = -2, the value of the given function takes the form  0/0 . 

13. Evaluate the Given limit  sin ax/bx 
Solution
 sin ax/bx 
At x = 0, the value of the given function takes the form 0/0 . 

14. Evaluate the Given limit : sin ax/sin bx, a,b ≠ 0

Solution

 (sin ax/sin bx), a,b ≠ 0
At x = 0, the value of the given function takes the form  0/0. 

15. Evaluate the Given limit  [sin(π -x)]/π(π - x) 
Solution

16. Evaluate the given limit  cos x/(π - x) 
Solution

17. Evaluate the Given limit :   (cos 2x - 1)/(cos x- 1) 
Solution
 (cos 2x - 1)/(cos x- 1)  
At x = 0, the value of the given function takes the form  0/0 .
Now,  

18. Evaluate the Given limit  (ax + x cos x)/b sin x
Solution
 (ax + x cos x)/b sin x 
At x = 0, the value of the given function takes the form 0/0. 
Now,  

19. Evaluate the Given limit  sec x 
Solution

20. Evaluate the Given limit  (sin ax + bx)/(ax + sin  bx) a,b, a+b ≠ 0.
Solution
At x = 0, the value of the given function takes the form 0/0 .
Now, 

21. Evaluate the Given limit  (cosec x - cot x)
Solution
At x = 0, the value of the given function takes the form ∞→∞.
Now, 

22. Evaluate the Given limit  
Solution

At x = π/2 , the value of the given function takes the form  0/0. 
Now, put x - π/2 = y so that x → π/2, y → 0. 

23. Find f(x)  and  f(x) where
Solution
The given function is 

24. Find  f(x) where 
Solution
The given function is  

25. Evaluate  f(x) where 
Solution

It is observed that  f(x) ≠  f(x).
Hence,  f(x) does not exist. 

26. Find  f(x) where 
Solution
The given function is  

It is observed that  f(x) ≠  f(x).
Hence,  f(x) does not exist .

27. Find  f(x) , where f(x) = |x| - 5
Solution
The given functions is f(x) = |x| - 5.

28. Suppose  and if  f(x) = f(1) what are possible values of a and b? 
Solution
The given function is  

f(x) =  (a + bx) = a + b
 f(x) = (b - ax) = b - a 
f(1) = 4 
It is given that  f(x) = f(1). 
∴  f(x) =  f(x) =  f(x) = f(1) 
⇒ a + b = 4 and b - a = 4 
On solving these two equations, we obtain a = 0 and b = 4. 
Thus, the respective possible values of a and b are 0 and 4.


29. Let a1, a2, . . ., an be fixed real numbers and define a function f ( x) = (x − a1) (x − a2)...(x − an). What is f(x) ? For some a ≠ a1, a2, ..., an, compute  
Solution
The given function is  f(x) = (x – a1)(x – a2) … (x - an). 

30. If   For what value (s)  of a does  f(x) exists ? 
Solution
The given function is  

When a = 0,

31. If the function f(x) satisfies  [f(x) - 2]/[x2 -1] = π , evaluate  f(x).
Solution

32. If 
For what integers m and n does 
 f(x) and  f(x) exist ? 
Solution
The given function is  
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